Inverse dynamic problems for canonical systems and de Branges spaces

A.S. Mikhaylov; V.S. Mikhaylov

REVIEW 3 major objections 5 minor 20 references

Inverse dynamic problems for canonical systems and de Branges spaces

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Inverse problems for wave, Dirac, and Jacobi systems reduce to canonical systems whose de Branges space is built from reachable states.

desk verdict A useful programmatic unification of dynamic inverse problems that derails on a concrete sign error in the key smooth-Hamiltonian reduction. read the letter →

arxiv 2505.23321 v1 pith:4WGK3WY6 submitted 2025-05-29 math.AP math.SP

A.S. Mikhaylov , V.S. Mikhaylov This is my paper

classification math.APmath.SP MSC 34A5534L4046E2247B36

keywords inverseproblemscanonicalsystemsdeBrangesspacesboundarycontrolmethodDiracsystemJacobimatriceswaveequationHamiltonian

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that inverse problems for four classical dynamical systems—the wave equation with a potential, the wave equation with a density, the Dirac system, and semi-infinite Jacobi matrices—are all equivalent to inverse problems for first-order canonical systems with a matrix Hamiltonian H. It exhibits explicit changes of variables that turn each system's boundary response operator into the response operator of a canonical system with one of three dynamics: second derivative, i d/dt, or a discrete difference. For canonical systems with a smooth strictly positive Hamiltonian, it outlines a boundary-control construction in which an extended control operator pairs the original system with an auxiliary system, the connecting operator is built from dynamic and spectral data, and the Fourier image of the extended reachable set becomes a de Branges space. If the construction works, the response data of any of the four systems determine the same Hamiltonian, making dynamic and spectral inverse data two views of one object. The general-Hamiltonian version is explicitly offered as a hypothesis.

What carries the argument

The central object is the Hamiltonian $H(x)$, a locally summable $2\times 2$ matrix-valued function with $H\ge 0$ and $\mathrm{tr}\,H=1$, appearing in the canonical system $iH \frac{dY}{dt} - J \frac{dY}{dx} = 0$ with $J$ the standard symplectic matrix. Each classical system is rewritten so that its boundary data become boundary data for such a canonical system. The argument then runs through the extended control operator $W^T$, which maps the extended control space $L^2(0,T;\mathbb{C}^2)$ to the state space by pairing solutions of the original system with solutions of an auxiliary system; the connecting operator $C^T = (W^T)^* W^T$ is the Gram matrix of this map. The de Branges space $B^T_D$ is defined as the Fourier image of the extended reachable set, with scalar product generated by $C^T$, where the Fourier transform uses the spectral measure of the associated Dirac-type operator.

What would settle it

Take a smooth strictly positive Hamiltonian and compute the norm of the extended control map $W^T$; if nonzero controls can produce arbitrarily small states, $W^T$ is not an isomorphism, $C^T$ is not positive definite, and the de Branges construction fails. For general Hamiltonians, a rank-changing Hamiltonian of Krein-string type is the natural place to look for such a counterexample.

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Extended reading notes

Core claim

The main claim is that the inverse problem for a canonical system $iH \frac{dY}{dt} - J \frac{dY}{dx} = 0$ carries the inverse problems for the wave equation with potential, the wave equation with density, the Dirac system, and Jacobi matrices: for each system the paper gives an explicit transformation of the unknown and the data after which the original response operator equals the canonical system's response operator. With the dynamics $\frac{d}{dt}$, smooth strictly positive $H$ yields finite wave speed, and the system reduces to a Dirac-type system. There the extended control operator $W^T$, built from the original and an auxiliary system, is stated to be an isomorphism; the connecting operator $C^T = (W^T)^* W^T$ is a positive operator expressible through the dynamic inverse data $R^{2T}$ and the spectral measure, and the Fourier image of the extended reachable set is a de Branges space. The paper presents this for smooth positive $H$ as a working scheme and for general $H$ as a hypothesis.

Load-bearing premise

The load-bearing premise is that every state at time $T$ can be reached by combined controls of the original and an auxiliary system, with the control norm equivalent to the state norm—a fact stated without proof, and for general Hamiltonians the construction is only a hypothesis.

Editorial extensions

If this is right

The response operator of any of the four systems—wave with potential, wave with density, Dirac, Jacobi—determines the same Hamiltonian $H$, so data collected for one system can in principle be reinterpreted for the others.
Dynamic inverse data $R^{2T}$ and spectral inverse data $d\rho$ both enter the connecting operator $C^T$, so the dynamic and spectral inverse problems are solved by one construction.
For smooth positive $H$, the de Branges space is obtained from reachable states at a fixed time, making the de Branges space a dynamic object rather than a purely spectral one.
For general $H$, solving the dynamic inverse problem is equivalent to dynamically constructing the de Branges space, so any method for one gives the other.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the isomorphism claim for $W^T$ fails only for some Hamiltonians, the natural repair is to replace the full state space by a weighted space; testing this on rank-changing Hamiltonians would clarify whether the de Branges construction survives without positivity.
The equivalence results imply that numerical solvers for any one of the four inverse problems can be ported to the others by transporting the Hamiltonian $H$; the paper does not draw this practical consequence.
Because the $i\frac{d}{dt}$ dynamics give finite propagation speed, the extended construction might adapt to multidimensional inverse problems in the style of the classical boundary-control method, where finite speed is essential; this is an extension, not a paper claim.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read the Mikhaylov-Mikhaylov note. The honest summary: Section 3 is a useful collection of explicit reductions of dynamic inverse problems for the wave equation (potential and density), the Dirac system, and Jacobi matrices to canonical systems with different time dynamics. Those transformations are mostly explicit and checkable, and the response-operator equivalences are genuinely useful for organizing the area. The paper is also candid about what is a hypothesis.

The soft spot is in Section 4.2, and it is not minor. The reduction of the canonical system (25) to the Dirac-type system (27) is algebraically wrong. Writing Y = U \tilde Y with U a rotation that diagonalizes H gives, after multiplying by U^*,

iD \tilde Y_t - J \tilde Y_x + φ' \tilde Y + i φ' D J \tilde Y = 0,

not the equation stated in (27). The paper has the sign of J d/dx flipped, the sign of φ' flipped, and it drops the extra off-diagonal term i φ' D J \tilde Y. That term cannot be absorbed into the scalar potential ψ of (28). So the whole Section 4.2 machinery, including Propositions 1–3 and the de Branges construction, is set up for a different equation than the one actually obtained from (25). The stress-test note is right, and the reader's independent worry about Proposition 2 stands; the more immediate problem is that the reduction feeding Proposition 2 is mis-signed.

I should also note: Propositions 1–3 are stated without proof. Proposition 2 (extended controllability) is load-bearing, and the paper itself acknowledges the original system is not boundary controllable. The general-Hamiltonian de Branges construction is explicitly a hypothesis. These would be acceptable in a genuine programmatic announcement if the transformation leading to them were correct, but here it is not.

Where does that leave the paper? The Section 3 content is worth having, and the overall idea—that these inverse problems can be unified through canonical systems—is reasonable. But the advertised strategy for smooth Hamiltonians does not yet hold together as written. The authors need to fix the transformation (maybe using a different gauge or absorbing the D J term into the potential) and supply proofs for the propositions.

Recommendation: send to peer review, but with a clear expectation of major revision. A serious referee will catch the sign error. If corrected, the paper could be a useful contribution; as written, it is a programmatic note with a load-bearing flaw.

Best,

Referee Report

3 major / 5 minor

Summary. The paper claims equivalences between dynamic inverse problems for the wave equation with potential, the wave equation with density, the Dirac system, and Jacobi matrices on the one hand, and inverse problems for first-order canonical systems with suitably chosen dynamics on the other. In Section 4, for a smooth strictly positive Hamiltonian, it proposes a boundary-control strategy for the canonical system, reduces it to a Dirac-type system, and outlines the construction of the associated de Branges space as the Fourier image of an extended reachable set. For a general Hamiltonian, the construction is formulated as a hypothesis rather than a theorem. The paper is explicitly an accompanying note to the authors' companion work [16] and defers several key arguments to forthcoming publications.

Significance. If the Section 3 equivalences hold, they provide a useful unifying framework for classical one-dimensional inverse problems, and the explicit response-operator identities are checkable and free of parameter fitting. The paper is honest that the Section 4 construction for general Hamiltonians is a hypothesis and identifies the two main obstacles, smoothness of H and changes of rank. However, the Section 4 claims rest on unproved propositions and contain an algebraic sign error in the reduction to the Dirac-type system, so the dynamic de Branges-space construction is not established as stated.

major comments (3)

[Section 4.2, Eq. (27)] The claimed reduction of (25) to a Dirac-type system is algebraically incorrect. Let U be a rotation with U^*HU=D. Since U^*JU=J and U^*JU_x=-\phi'I, substituting Y=U\tilde Y into iH Y_t - JY_x=0 gives iD\tilde Y_t - J\tilde Y_x + \phi'\tilde Y=0, not iD\tilde Y_t + J\tilde Y_x - \phi'\tilde Y=0 as stated in (27). The error is visible already for \phi\equiv 0, where (27) would claim the opposite sign of the spatial derivative. The boundary conditions in the same paragraph are also inconsistent: the correct formulas are \tilde y_1(0,t)=\cos\phi(0)f(t)-\sin\phi(0)(Rf)(t) and \tilde y_2(0,t)=\sin\phi(0)f(t)+\cos\phi(0)(Rf)(t), not the expressions with the \sin terms interchanged. Because the auxiliary system (29), the extended control operator W^T, and Proposition 2 are all defined relative to (28), the entire Section 4.2 construction applies to a dynamics different from the one actually equivalent to (25). This is load-bearing for the de Branges construction in Section 4 and must be corrected.
[Section 4.2, Propositions 1-3] The three propositions are essential to the dynamic de Branges construction and are stated without proof. In particular, Proposition 2 asserts that the extended control operator W^T: L^2(0,T;C^2) -> L^2(0,\tau(T);C) is an isomorphism. This is what guarantees positivity of the connecting operator C^T and hence that the inner product on the Fourier image B^T_D is positive definite. The paper notes that the original system (28) is not boundary controllable, so the auxiliary system (29) must restore controllability, but no argument for surjectivity or boundedness below of W^T is provided. Proposition 3, the representation of C^T in terms of R^{2T} and the spectral measure d\rho, is likewise asserted without derivation. Since the dynamic de Branges-space construction depends on these facts, the central claim of Section 4 is not established as a theorem as written; at minimum, the propositions should be proved or explicitly reformulated as conjectures with precise hypotheses.
[Section 4.2, relation between (28) and (29)] The statement that solutions to (28) and (29) are connected by V^f = U^f is not consistent with the signs of the equations. If V solves (28), then U(x,t)=V(x,-t) solves (29), not U=V. This matters because the extended control operator W^T is defined by summing V^f(x,T) and U^g(x,T), so a time reversal in the auxiliary channel changes the reachable set and the resulting connecting operator C^T. This point should be corrected together with the sign error in (27).

minor comments (5)

[Section 3.2] After defining C(x,t), the text says 'Y satisfies the canonical system (9)', but the symbol should be C, not Y.
[Section 3.1] The response operator for (7) is denoted \tilde R^T_s, but the subscript should be q; the same subsection uses \tilde R^T_s f inconsistently.
[Section 3.4] In equation (20) the equation number is inserted inside the displayed formula, making the expression hard to read.
[Section 4.2, Proposition 1] In the integral representation, the integration variable is s but the upper limit is written x(t); since x(t) is the inverse of \tau(x), the notation should be made consistent with the region 0\le \tau(s)\le t.
[Section 4.1] The domain of R^T_w is written as 'L^2(0,T;C)7\rightarrow L^2(0,T;C)' with stray symbols, and the phrase 'is introduced as ,' contains an extra comma.

Circularity Check

1 steps flagged · score 3.0 of 10

The Section 3 equivalences are genuine self-contained reductions, but the Section 4.2 de Branges/BC construction rests on Propositions 2-3, which are asserted without proof and sourced to the authors' own companion paper [16]; this makes the central new construction partially self-citation-load-bearing, though no fitting-as-prediction circularity is present.

self citation load bearing [Section 4.2, Propositions 2 and 3 (after Eqs. (29)-(31)); cf. Section 4.3]
"We outline the scheme offered in [9, 16]: ... Proposition 2. The extended control operator is an isomorphism between F T and H T . ... Proposition 3. The extending connecting operator is a positive isomorphism in F T , it admits the representation in terms of dynamic inverse data R 2T , and spectral inverse data dρ(λ)."

The proposed solution of the dynamic inverse problem and the construction of the de Branges space depend entirely on Propositions 2 and 3: positivity and response-data computability of C^T are what convert the reachable set into a usable de Branges space. Neither proposition is proved here; the text only cites the authors' own [16] (and [9], co-authored by one of the authors) as the source of the scheme. Thus the paper's central Section 4 construction is not independently derived in this manuscript; it reduces to an unverified-in-text self-citation, with further proof deferred to 'forthcoming publications.' This is load-bearing, though not a fit-as-prediction circularity.

full rationale

No fitted parameters are introduced and no response datum is renamed as a prediction; the Section 3 reductions (wave with potential to (7), density wave to (9), Dirac to (12), Jacobi to (22)/(24)) proceed algebraically from the stated equations and are self-contained. The score is elevated only by Section 4.2-4.3: the core de Branges construction for a general Hamiltonian is presented as an outline based on Propositions 2-3, which are asserted without proof and attributed to the same authors' companion paper [16]. This is a self-citation used as the load-bearing justification rather than a re-derivation, so a moderate score is appropriate. The skeptic's algebraic-sign objection to the reduction from (25) to (27) is a correctness concern, not a circularity, and is therefore not counted in the score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical parameters are fitted. The listed axioms are mostly standard background, plus several unproved controllability and representation assumptions that carry the Section 4 construction. The paper introduces no new physical entities; its new objects are mathematical (extended control operator, dynamic de Branges space), and they are explicitly conditional on the theorems being proved.

assumptions (6)

standard math The standard theory of de Branges spaces and canonical systems from [17, 18, 10] is assumed, including the equivalence between Hermite-Biehler functions and canonical systems.
Section 2 states the de Branges space setup and cites [17,18,10] for the inverse spectral theorem; the rest of the paper relies on this background.
domain assumption In Section 4, the Hamiltonian is assumed smooth, strictly positive, self-adjoint, with trH=1.
Section 4 opening: 'Assuming that the Hamiltonian satisfies conditions: H=H* in C^2(0,T;R^{2x2}), H >= delta > 0, trH=1'. This excludes the singular rank-one and piecewise constant Hamiltonians that appear in Section 3.
ad hoc to paper The solution representation in Proposition 1 exists with a twice differentiable kernel w and amplitude A solving the stated ODE system.
Proposition 1 in Section 4.2 is stated without proof; the entire controllability analysis depends on this representation.
ad hoc to paper The extended control operator W^T in Proposition 2 is an isomorphism.
Proposition 2, Section 4.2: 'The extended control operator is an isomorphism between F^T and H^T'. This is the weakest assumption and is not proved in the preprint.
ad hoc to paper The connecting operator C^T admits a representation in terms of the dynamic inverse data R^{2T} and the spectral measure d rho.
Proposition 3, Section 4.2, is stated without proof and is the step that connects boundary measurements to the de Branges space metric.
ad hoc to paper For general Hamiltonian, the dynamic de Branges space can be built as the Fourier image of the extended reachable set with the C^T metric.
Section 4.3 states this as a hypothesis, and the authors note obstacles from smoothness and rank changes of H.

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Cite this review

Pith. "Pith review of Inverse dynamic problems for canonical systems and de Branges spaces." pith.science (2026). https://pith.science/paper/4WGK3WY6

@misc{pith2026250523321,
  author       = {Pith},
  title        = {Pith review of: Inverse dynamic problems for canonical systems and de Branges spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4WGK3WY6}},
  note         = {Machine review of arXiv:2505.23321}
}

read the original abstract

We show the equivalence of inverse problems for different dynamical systems and corresponding canonical systems. For canonical system with general Hamiltonian we outline the strategy of studying the dynamic inverse problem and procedure of construction of corresponding de Branges space.

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Works this paper leans on

20 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [16]

A. S. Mikhaylov, V. S Mikhaylov . Dynamic inverse problem for the Jacobi matrices. https://arxiv.org/abs/1704.02481

work page Pith review arXiv

[2] [1]

S. A. Avdonin, S. Lenhart and V. Protopopescu. Determining the potential in the Schr\" o dinger equation from the Dirichlet to Neumann map by the boundary control method J. Inverse Ill-Posed Probl, 13, 2005, 317--330

work page 2005

[3] [2]

S. A. Avdonin, V. S. Mikhaylov . The boundary control approach to inverse spectral theory. Inverse Problems, 26, no. 4, 2010, 045009, 19 pp

work page 2010

[4] [3]

M. I. Belishev. An approach to multidimensional inverse problems for the wave equation Dokl. Akad. Nauk SSSR, 297, no. 3, 1987, 524--527 (in Russian) M. I. Belishev. Soviet Math. Dokl. 36, no. 3, 1988, 481--484 (Engl. Transl.)

work page 1987

[5] [4]

M. I. Belishev. Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Problems 13, no. 5, 1997, R1--R45

work page 1997

[6] [5]

M. I. Belishev. Boundary control and inverse problems: a one-dimensional version of the boundary control method. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 354, 2008, 19--80 (in Russian); English translation: J. Math. Sci. (N. Y.), 155, no 3, 2008, 343-v378

work page 2008

[7] [6]

Recent progress in the boundary control method

M.I.Belishev . Recent progress in the boundary control method. Inverse Problems, 23, no 5, 2007, R1--R67

work page 2007

[8] [7]

Boundary control and tomography of Riemannian manifolds (the BC-method)

M.I.Belishev. Boundary control and tomography of Riemannian manifolds (the BC-method). Uspekhi Matem. Nauk, 72, no 4, 2017, 3-66 (in Russian)

work page 2017